Hamiltonian-with-Handles Graphs and the k-Spine Drawability Problem

Abstract

A planar graph G is k-spine drawable, k≥ 0, if there exists a planar drawing of G in which each vertex of G lies on one of k horizontal lines, and each edge of G is drawn as a polyline consisting of at most two line segments. In this paper we: (i) Introduce the notion of hamiltonian-with-handles graphs and show that a planar graph is 2-spine drawable if and only if it is hamiltonian-with-handles. (ii) Give examples of planar graphs that are/are not 2-spine drawable and present linear-time drawing techniques for those that are 2-spine drawable. (iii) Prove that deciding whether or not a planar graph is 2-spine drawable is $\mathcal{NP}$-Complete. (iv) Extend the study to k-spine drawings for k >2, provide examples of non-drawable planar graphs, and show that the k-drawability problem remains $\mathcal{NP}$-Complete for each fixed k > 2.